Equations of motion for polarization moments

As has already been mentioned in Section 3.1, the simplest way of accounting for the symmetry properties of processes participating in optical pumping is the expansion of the density matrices /mm and over irreducible tensor operators Tq [136, 140, 304, 379]  

The choice of the phase and the normalization of the irreducible tensor operators is somewhat arbitrary [379]. Following [133], we will employ the following definition of matrix elements of tensor operators (other existing methods are discussed in Appendix D)  

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

Applying the Wigner-Eckart theorem [136, 140, 304, 379] to the matrix elements of form (M E d /z) (see (5.9)), the latter may be written through 

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With the exception of the orientation factor, all the parameters in this equation may be obtained within reasonable error by direct experimental measurement or by estimation. The problem of setting reasonable values for k2, which may vary from 0 to 4 for orientations in which the dipole moments are orthogonal or parallel, respectively, is nontrivial. A value of , which is an unweighted average over all orientations, is often used. Dale et al.(53) have examined this problem in great detail and have shown that a k2 value of is never justified for energy transfer in macromolecules because it is impossible for the donors and acceptors to achieve a truly isotropic distribution. They do provide an experimental approach, using polarized emission spectroscopy, to estimate the relative freedom of motion for the donor and acceptor that allows reasonable limits to be set for k2. 

In order to obtain the equations of motion of classical polarization moments, we must base our methods on the system of equations of motion of the probability density pa(6, angular momentum vector 3(6, optical pumping. For a number of maximally simplified situations, where the probability density in the ground state pa(6,(p) does not depend on that of the excited state pb(6,(p), we have already encountered such equations in preceding chapters see e.g., (3.4), or (4.5) and (4.6). 

Although the Bloch vector-rotating frame formalism is convenient for deriving the basic equation of motion and for understanding the similarities among transient phenomena in various fields, we find it more convenient to work with physically real quantities when describing specific experiments. The physical quantities of interest in microwave experiments are the polarization, or macroscopic induced dipole moment, per unit volume, and the population difference between the levels a and b per unit volume.1... 

Let us consider the wave vectors K from the vicinity of the first Brillouin zone origin (i.e. K 0). It corresponds to the infrared active lattice vibrations with X > > a. The optical phonon branch has the highest vibration frequency possible for the atom chain in that case. The ions vibrate with the opposite phases and amplitudes inversely proportional to their masses. Dipole moments are effectively created in each elementary unit and therefore the crystal is polarized. Polarization of the crystal causes the internal periodical electric field E at the position of each atom. This field contributes to the additional electric force on each ion, either by - -eEi, or by —eE force. Let us further denote the stiffness of the nearest neighbor interaction (i.e. spring constant) by C and displacement of ions A and B by w.4, ub respectively. The ion displacements follow the differential equation of motion... 

Due to the high polarity of these polymers the location of the fluorine atoms in the aromatic ring play an important role on the molecular motions below glass-rubber transition. For this reason the knowledge of the mean square dipole moment per repeating unit, (/u2)/x, which is calculated by means of the Guggenheim- Smith equation [173-175] ... 

This is the Langevin equation which describes the degree of polarization in a sample when an electric field, E, is applied at temperature T. Experimentally, a poling temperature in the vicinity of Tg is used to maximize dipole motion. The maximum electric field which may be applied, typically 100 MV/m, is determined by the dielectric breakdown strength of the polymer. For amorphous polymers p E / kT 1, which places these systems well within the linear region of the Langevin function. The following linear equation for the remanent polarization results when the Clausius Mossotti equation is used to relate the dielectric constant to the dipole moment 41). 

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